Derivation

Split input into 2 regimes
if (- (log (+ N 1)) (log N)) < 8.96628332219472e-07
1. Initial program 59.9
  \[\log \left(N + 1\right) - \log N\]
2. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^{2}}}\]
3. Simplified0.0
  \[\leadsto \color{blue}{(\left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) \cdot \left(\frac{1}{N \cdot N}\right) + \left(\frac{1}{N}\right))_*}\]
if 8.96628332219472e-07 < (- (log (+ N 1)) (log N))
1. Initial program 0.2
  \[\log \left(N + 1\right) - \log N\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.2
  \[\leadsto \log \color{blue}{\left(\sqrt{N + 1} \cdot \sqrt{N + 1}\right)} - \log N\]
4. Applied log-prod0.2
  \[\leadsto \color{blue}{\left(\log \left(\sqrt{N + 1}\right) + \log \left(\sqrt{N + 1}\right)\right)} - \log N\]
5. Applied associate--l+0.2
  \[\leadsto \color{blue}{\log \left(\sqrt{N + 1}\right) + \left(\log \left(\sqrt{N + 1}\right) - \log N\right)}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 8.96628332219472 \cdot 10^{-07}:\\ \;\;\;\;(\left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) \cdot \left(\frac{1}{N \cdot N}\right) + \left(\frac{1}{N}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + N}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log N\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018217 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))